It's a rule of thumb, not an iron-clad law.

For the tax year 2023, you can contribute up to $6500 of earned income (meaning, income that results from a job) into your Roth IRA. Once you reach that limit, you are done contributing to this year's Roth IRA. Fortunately, the contribution counter starts fresh each year. If you have additional money you want to invest, you could do it in a non-tax-advantaged account like a brokerage account.

Given your current living situation, it's probably not imperative that you set aside a 3-6 month emergency fund, but it also wouldn't be a bad idea to have 2-3 months set aside, just in case something really unexpected comes up.

My Marcus by Goldman Sachs HYSA is currently getting 3.75% APY. It isn't 100% liquid, but it only takes a day or two to transfer it back to my checking account should I need it.